AD-Ai5 468 HARTREE-FOCK INTERACTION BETWEEN A HELIUM ATOM AND i,'LITHIUM-METAL: A TEST F.. (U) CORNELL UNIV ITHACA NY LABOF ATOMIC AND SOLID STATE PHYSICS.. B K RAO ET AL.

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~artree-Fock interaction between a helium Atom andItemLithium-metal: A test for the effective Medium _______________

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4 EN TARY NOTES

to Physical Review Letters.

CS3 (Continu, on roer.,. .d If an.em d Identify by block number)

22. A 9STPACT (Continue on -&-*~o ad*e itn.co.#'7 Mid identllf by block number)

The interaction of a helium atom with 6 and 10 atom clusters of lithium has beei* calculated using the unrestricted Hartree-Fock method, Hartree-Fock method with

correlation corrections, and the effective medium theory. Inside the cluster thehelium embedding energy is found to be proportional to the electron density of thecluster. The proportionality constant obtained by the Hartree-Fock method is in faiIagreement with that calculated in the homogeneous electron gas using the local densiy~approximation. Outside the cluster, in the region compatiblb with the helium diffra -

Ition experiments, the self-consistent calculations give much larger repulsion thanD 0 , 1473 ECITION 01 I NOV 63 15 OaSOt.ETE

srcu-wrT!2 51.TSICAIO OF T.15 PAGI tI ^%0p

Hartree-Fock interaction between a helium atom

and lithium metal: a test for the effective medium theory

B.K. Rao and P. JenaDepartment of PhysicsVi rgi ni a Commonweal th

University, Richmond, VA 23284

D. D. SchilladyDepartment of ChemistryVirginia Commonwealth

University, Richmond, VA 23284

A. HlntermannLaboratory of Medium Energy Physics.-Swiss Institute for Nuclear Research -

CH-5234 Villigen, Switzerland

M. ManninenLaboratory of Atomic and Solid State Physics

Cornell UniversityIthaca, NY 14853

Accession For

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Abstract

-)The interaction of a helium atom with 6 and 10 atom clusters of

lithium has been calculated using the unrestricted Hartree-Fock method,

Hartree-Fock method with correlation corrections, and the effective medium -

theory. Inside the cluster the helium embedding energy is found to be

proportional to the electron density of the cluster. The proportionality

constant obtained by the Hartree-Fock method is in fair agreement with -

that calculated in the homogeneous electron gas using the local density

approximation. Outside the cluster, in the region compatible with the

helium diffraction experiments, the self-consistent calculations give much

larger repulsion than the effective medium theory.,'

7. C

.PL '-.

The utilization of helium diffraction for detailed structural

studies of surfaces requires a method for linking the scattering potential

to the structure. A fully self-consistent calculation of the potential

energy of the helium atom outside a semi-infinite metal surface has been

so far, too complicated to accomplish but several theoretical or

semiempirical approaches have been suggested for determining the

scattering potential 1-7.- All these approaches give the result that the

helium potential energy is roughly proportional to the unperturbed

electron density at the surface. This relationship comes as a natural

result of the so-called effective medium8 or quasiatom9 theory which

calculates the impurity binding energy as a functional of the electron

density of the host. The leading repulsive term of the scattering

potential can be written as6'10

a eff5'( (1

where aeff is a constant and ~()is the average of the host electron

density over the electrostatic potential of the helium atom. It turns out

that outside a metal surface as well as inside the metal A(ir) is, within a

good occuracy, proportional to n(h), the local host electron density at

the helium site. Then approximately

VO) ;ffn~h(2)

where ff is a new constant.6

The effective medium theory has been tested by model calculations.

The helium trappling energies in jellium vacancies, and the helium

potential energy outside a jellium surface calculated by the effective

%-A % -%~%%~ %%*~*so far,.oo coplicate.to*acomplis butsvera theoretical or-'-*-.

3

medium theory agree well with fully self-consistent calculations. 9'°10

Also, the calculated helium trapping energies in real metal vacancies

agree well with experimental results 1,12. When applied to helium dif-

fraction the effective medium theory gives qualitatively the correct

features of the potential 2 but fails to give the corrugation of the poten-

tial quantitatively correctly.-

In this letter we report on self-consistent Hartree-Fock cluster

calculations for He-Li interaction and compare the results with those of the

effective medium theory. We show that inside the metal the effective medium

theory gives good results whereas outside the cluster, in the region where

the helium scattering takes place, the self-consistent Hartree-Fock potential

is more repulsive than the prediction of the effective medium theory.

The two Li clusters considered are shown in Fig. 1. The helium

atom was moved from the octahedral interstitial site outside the cluster

as Indicated in the figure, and self-consistent total energy calculations

were performed for several helium sites along that line. The computations

for the cluster with 6 lithium atoms were repeated using several different

sets of basis functions;",' s to make sure that the choice of basis sets

did not introduce any uncertainties. For testing the sensitivity of the

results on the cluster size a calculation has also been made for a cluster

with ten lithium atoms. In the best calculation using the 6-31G basis 14 ,15

the effect of correlation was included by a perturbative technique (so

called MP2) 19 ,2 °. Using the same basis set we also calculated the He-He

potential and found a good agreement with earlier calculations. 6

The results of the self-consistent calculations are shown in Fig. 2

together with the result of the effective medium theory, Eq. (1). The

x *- . ot°

4 '

results computed using different basis sets show that inside the cluster

the helium energy depends slightly on the basis set used whereas outside

the cluster the difference becomes smaller. The correlation correction

lowers the helium energy by about 2% in the center of the cluster and at

the distance of 8 a.u. about 7%. The cluster calculations of the impurity

1 p..energy often show sensitivity on the clusters size ; 7 and geometry18 . To

check the dependence of the helium energy on the cluster size we have

repeated one calculation by adding 4 Li atoms in the 6 atomic cluster as -

shown in Fig. 1. These additional atoms change the helium potential

inside the cluster but do not have a noticeable effect outside the

cluster.

The result of the effective medium theory was calculated using the

self-consistent Hartree-Fock electron density (Li 4-31q basis set) and the

coefficient 6 aeff=196eV/ao 3 in Eq. (1). In the case of Li since the

averaged density n() is within 3$ of the local density n( in all helium

sites, we can use approximation

aeff " aeff. (3)

which is valid only for Li and not necessarily for any other metal.

Figure 2 shows that the effective medium theory describes the interaction

of the helium atom within the Li cluster qualitatively correctly.

Using Eq. (2) we can determine the coefficient -eff once the

self-consistent total energy and electron density are known. In Fig. 3

oeff determined from the Hartree-Fock results is shown as a function of

the helium position. The effective medium theory predicts that -eff is

nearly constant. The Hartree-Fock result for -eff is nearly constant

A..-.

., ,- - -, , .. .• . . . .. ... . . .. .. ..... .. . .X _ _ .--.,

.%1 .'°W.~~~ -707 o7 7-

inside the metal and only slightly larger than that obtained for a

homogeneous electron gas. However, outside the cluster where the electronI

density becomes small aeff increases rapidly. At a helium distance of 8

au. which corresponds to a typical helium scattering distance, the

Hartree-Fock calculation already gives 3 times more repulsive potential

than the effective medium theory. This is in accordance with the experi-

mental findings that the actual scattering potential is much less corru-

gated than the result of the effective medium theory 5,13, suggesting S

that the scattering occurs further from the surface.

There are two possibilities why the effective medium theory is

incapable of describing correctly the repulsion at large distances:

either the nonlocal treatment of the exchange-correlation energy becomes

important or the higher order corrections of the effective medium theory

can not be neglected. The model calculation of Lang and Norskov10 for

jellium seems to indicate that the higher order corrections are small, but

this is not necessarily the case at actual metal surfaces where the

Freidel oscillations induced by the helium atom will overlap with the

strong ion core potentials. In any case, it seems that the effective

medium theory in its present form is incapable of calculating quantita-

tively correctly the helium potential in the region relevant to theI;diffraction experiments.

In conclusion, we have performed unrestricted Hartree-Foch calcula-

tions (including correlation corrections) for the interaction potential

between a helium atom and a 6 atom clfster of lithium. Inside the cluster

the potential is roughly proportional' to the self-consistent electron den-

sity of the lithium cluster in accordance with the effective medium

theory, but outside the cluster the potential is much more repulsive thanpredicted by the effective medium theory.

6

Acknowledgements

Discussions with J. K. Norskov, T. Rantala, C. lUmrigar and J. W.

Wilkins are gratefully acknowledged. This work was supported in part by

grants from Thomas F. Jeffress and Kate Miller Jeffress Foundation,

National Science Foundation, Academy of Finland, and by The Petroleum

Research Fund, administered by the ACS.

Ov5-

J~ %' ~ T ~ - . . - -- * '.

1.7

References

1. E. Zaremba and W. Kohn, Phys. Rev. B 15, 1769 (1977).

2. M. Manninen, J. K. Norskov and C. Umrigar, Surf. Sct. 119, L393

(1982).

3. J. Harris and A. Liebsch, J. Phys. C 15, 2275 (1982).

4. R. B. Laughlin, Phys Rev. B 25, 2222 (1982).

5. R. Schinke and A. C. Luntz, Surf. Sc. 124, L60 (1983).

6. M. Manninen, J. K. Norskov, M. J. Puska and C. Umrigar, Phys. Rev. B

29, 2314 (1984). The proportionality coefficient a published in the

original paper of Esbjerg and Norskov (Phys. Rev. Lett. 45, 807

(1980)) was numerically erroneous.

7. J. F. Annet and R. Haydock, Phys. Rev. Lett. 53, 838 (1984).

8. J. K. Norskov and N. D. Lang, Phys. Rev. B 21, 2131 (1980).

9. M. J. Stott and E. Zaremba, Phys. Rev. B 22, 1564 (1980).

10. N. D. Lang and J. K. Norskov, Phys. Rev. B 27, 4612 (1983).

11. M. Manninen, J. K. Norskov and C. Umrigar, J. Phys. F 12, 17 (1982).

12. M. J. Puska and R. M. Nieminen, Phys. Rev. B 29, 5382 (1984).

13. M. Karikorpi, M. Manninen and C. Umrigar, to be published.

14. For Li atom STO-3G, 4-31G and 6-31G were used. For the He atom we

added optimized 2s and 2p functions to the original basis set.

15. W.J. Hehre, R. F. Steward, and J. A. Pople, J. Chem. Phys. 51, 2657

(1969); W. J. Hehre, R. Oitchfield, R. F. Steward, and J. A. Pople,

J. Chem. Phys. 52, 2769 (1970).

16. P. E. Phillipson, Phys. Rev. 125, 1981 (1962), H. F. Schoefer, D. R.

McLaughlin, F. E. Harris, B. J. Alder, Phys. Rev. Lett. 25, 988

(1970).

:**..-L- :

.. F ? - o o F F - - ° -o . o o .... °, -o

° • F° " -l . .. • • • ° °

° -.

17. A. Hintermann and M. Manninen, Phys. Rev. B 21, 7267 (1983).

18. B. K. Rao, P. Jena and M. Manninen, to be published.

19. C. M'oller and M. S. Plesset, Phys. Rev. 46, 618 (1934).

20. J. A. Pople, R. Krishnan, H. B. Schlegel, and J. S. Binkley, Int. J.

Quant. Chem. 14, 545 (1978).

9

Figure Captions

Figure 1. Structure of the 6 and 10 atomic lithium clusters considered.

Black dots show the 6 atomic BCC octahedron and the open circles the

additional atoms in the 10 atomic cluster. The helium atom is moved along

the heavy solid line.

Figure 2. Potentials between the He atom and the 6 and 10 atomic LiS

clusters. The Li atom basic set is denoted for each curve. EMT is the

result of the effective medium theory.

Figure 3. The proportionality coefficient eff of Eq. (2) determined k,

from the Hartree-Fock result (4-31g basis) EMT is the result calculated

for a homogeneous electron gas6.

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